This proofs derived from Schauder-Tychonov fixed point Theorem (see, for example, [1]}, [2]}).
Let \(n\)  in \(\mathbb {N}\)  be fixed, let \(v\)  be a function in \(L^{p(.)}(\Omega ),\)  we know that the following non singular problem
\(\left\lbrace \begin{array}{cl}-\operatorname{div}\left(a(x)|\nabla w|^{p(x)-2} \nabla w\right)+b(x)|w|^{r(x)-1} w=\frac{f_{n}}{\left(|v|+\frac{1}{n}\right)^{\gamma (x)}} & \text{ in } \Omega \\w=0 & \text{ on } \partial \Omega .\end{array}\right.\) 
